Problem: A complex number $z_1$ has a magnitude $|z_1|=7$ and an angle $\theta_1=270^{\circ}$. Express $z_1$ in rectangular form, as $z_1=a+bi$. Express $a+bi$ in exact terms. $z_1 = $
The Strategy A complex number of the form $z={a}+{b}i$ has: A magnitude of ${|z|}=\sqrt{{a}^2+{b}^2}$. An angle of ${\theta}=\arctan\left(\dfrac{{b}}{{a}}\right)$. [How did we get these equations?] Therefore, given the absolute value ${|z|}$ and angle ${\theta}$, the parts ${a}$ and ${b}$ are given by the following two equations: ${a}={|z|}\cos{\theta}$ ${b}={|z|}\sin{\theta}$ [How did we get these equations?] Finding $a$ For ${|z_1|}={7}$ and ${\theta_1}={270^{\circ}}$, we can find ${a}$ as follows. $\begin{aligned}{a}&={|z_1|}\cos{\theta_1} \\\\&={7}\cos{270^\circ} \\\\&={0}\end{aligned}$ Finding $b$ $\begin{aligned}{b}&={|z_1|}\sin{\theta_1} \\\\&={7}\sin{270^\circ} \\\\&={-7}\end{aligned}$ Summary We found that ${a}={0}$ and ${b}={-7}$. Therefore, $z_1=0-7i$.